Problem: As the swim coach at Springer, Ishaan selects which athletes will participate in the state-wide swim relay. The relay team swims $\frac{5}{7}$ of a mile in total, with each team member responsible for swimming $\frac{5}{14}$ of a mile. The team must complete the swim in $\frac{3}{4}$ of an hour. How many swimmers does Ishaan need on the relay team?
Solution: To find out how many swimmers Ishaan needs on the team, divide the total distance ( $\frac{5}{7}$ of a mile) by the distance each team member will swim ( $\frac{5}{14}$ of a mile). $ \dfrac{{\dfrac{5}{7} \text{ mile}}} {{\dfrac{5}{14} \text{ mile per swimmer}}} = {\text{ number of swimmers}} $ Dividing by a fraction is the same as multiplying by the reciprocal. The reciprocal of ${\dfrac{5}{14} \text{ mile per swimmer}}$ is ${\dfrac{14}{5} \text{ swimmers per mile}}$ $ {\dfrac{5}{7}\text{ mile}} \times {\dfrac{14}{5} \text{ swimmers per mile}} = {\text{ number of swimmers}} $ $ \dfrac{{5} \cdot {14}} {{7} \cdot {5}} = {\text{ number of swimmers}} $ Reduce terms with common factors by dividing the $5$ in the numerator and the $5$ in the denominator by $5$ $ \dfrac{{\cancel{5}^{1}} \cdot {14}} {{7} \cdot {\cancel{5}^{1}}} = {\text{ number of swimmers}} $ Reduce terms with common factors by dividing the $14$ in the numerator and the $7$ in the denominator by $7$ $ \dfrac{{1} \cdot {\cancel{14}^{2}}} {{\cancel{7}^{1}} \cdot {1}} = {\text{ number of swimmers}} $ Simplify: $ \dfrac{{1} \cdot {2}} {{1} \cdot {1}} = {2} $ Ishaan needs 2 swimmers on his team.